We develop a general theory of transport-limited aggregation phenomena occurring on curved surfaces, based on stochastic iterated conformal maps and conformal projections to the complex plane. To illustrate the theory, we use stereographic projections to simulate diffusionlimited-aggregation (DLA) on surfaces of constant Gaussian curvature, including the sphere (K > 0) and pseudo-sphere (K < 0)...

A conformal mapping in a plane domain locally maps circles to circles. More generally, quasiconformal mappings locally map circles to ellipses of bounded distortion. In this work, we study the corresponding situation for solutions to Stein-Weiss systems in the (n + 1)D Euclidean space. This class of solutions coincides with the subset of monogenic quasiconformal mappings with nonvanishing hyper...

We introduce the notion of a conformal de Rham complex of a Riemannian manifold. This is a graded differential Banach algebra and it is invariant under quasiconformal maps, in particular the associated cohomology is a new quasiconformal invariant.

The remarkable theorem of P. Belinskii is a deep underlying result in the variational calculus for quasiconformal maps. It valid only maps with small sufficiently regular Beltrami coefficients. We provide global version this connected classical results on extensions conformal and its new applications.

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this ex...

Diffusion-limited aggregation (DLA) and its variants provide the simplest models of fractal patterns, such as colloidal clusters, electrodeposits, and lightning strikes. The original model involves random walkers sticking to a growing cluster, but recently DLA (in the plane) has been reformulated in terms of stochastic conformal maps. This fruitful new perspective provides the exact Laplacian c...

Accurately simulating fluid dynamics on arbitrary surfaces is of significance in graphics, digital entertainment, and engineering applications. This paper aims to improve the efficiency and enhance interactivity of the simulation without sacrificing its accuracy. We develop a GPU-based fluid solver that is applicable for curved geometry. We resort to the conformal (i.e., angle-preserving) struc...

The previously analyzed holographic encoding of bulk matter is generalized from wedges to double cones. As a result of the conformal invariance of the holographically projected wedge-bulk matter, one may apply a conformal transformation in the ambient space which maps the holographic projection of the wedge into that of the double cone. In the massive case this conformal map cannot be used for ...

We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane H, say) which satisfy the “conformal restriction” property, i.e., K connects two fixed boundary points (0 and ∞, say) and the law of K conditioned to remain in a simply connected open subsetH of H is identical to that of Φ(K), where Φ is a conformal map from H onto H with Φ(0) = 0 a...

Accurately simulating fluid dynamics on arbitrary surfaces is of significance in graphics, digital entertainment, and engineering applications. This paper aims to improve the efficiency and enhance interactivity of the simulation without sacrificing its accuracy. We develop a GPU-based fluid solver that is applicable for curved geometry. We resort to the conformal (i.e., angle-preserving) struc...